Question

If the lines 2x-3y+5=0, 9x-5y+14=0 and 3x-7y+λ=0 are concurrent, then the value of λ is equal to

• A. 7
• B. 8
• C. 10
• D. 9
• E. 6

Answer 1

The Correct Option is C

For three lines to be concurrent, the determinant of their coefficients must be equal to 0. The given lines are: \[ 2x - 3y + 5 = 0 \quad \text{(line 1)} \] \[ 9x - 5y + 14 = 0 \quad \text{(line 2)} \] \[ 3x - 7y + \lambda = 0 \quad \text{(line 3)} \] The system of equations for the lines is: \[ \begin{vmatrix} 2 & -3 & 5 \\ 9 & -5 & 14 \\ 3 & -7 & \lambda \end{vmatrix} = 0 \] We will now calculate the determinant: \[ \text{Determinant} = 2 \begin{vmatrix} -5 & 14 \\ -7 & \lambda \end{vmatrix} - (-3) \begin{vmatrix} 9 & 14 \\ 3 & \lambda \end{vmatrix} + 5 \begin{vmatrix} 9 & -5 \\ 3 & -7 \end{vmatrix} \] Expanding each of the 2x2 determinants: \[ \begin{vmatrix} -5 & 14 \\ -7 & \lambda \end{vmatrix} = (-5)(\lambda) - (14)(-7) = -5\lambda + 98 \] \[ \begin{vmatrix} 9 & 14 \\ 3 & \lambda \end{vmatrix} = (9)(\lambda) - (14)(3) = 9\lambda - 42 \] \[ \begin{vmatrix} 9 & -5 \\ 3 & -7 \end{vmatrix} = (9)(-7) - (-5)(3) = -63 + 15 = -48 \] Substituting these values back: \[ \text{Determinant} = 2(-5\lambda + 98) + 3(9\lambda - 42) + 5(-48) \] \[ = -10\lambda + 196 + 27\lambda - 126 - 240 \] \[ = 17\lambda - 170 \] For the lines to be concurrent, the determinant must be 0: \[ 17\lambda - 170 = 0 \] Solving for \( \lambda \): \[ 17\lambda = 170 \] \[ \lambda = 10 \]

The correct option is (C) : \(10\)

Answer 2

The lines 2x - 3y + 5 = 0, 9x - 5y + 14 = 0, and 3x - 7y + \(\lambda\) = 0 are concurrent if and only if the determinant of their coefficients is zero.

That is, the following determinant must be zero:

\(\begin{vmatrix} 2 & -3 & 5 \\ 9 & -5 & 14 \\ 3 & -7 & \lambda \end{vmatrix} = 0\)

Expanding the determinant, we have:

\(2 \begin{vmatrix} -5 & 14 \\ -7 & \lambda \end{vmatrix} - (-3) \begin{vmatrix} 9 & 14 \\ 3 & \lambda \end{vmatrix} + 5 \begin{vmatrix} 9 & -5 \\ 3 & -7 \end{vmatrix} = 0\)

This gives us:

\(2(-5\lambda - (-7)(14)) + 3(9\lambda - (3)(14)) + 5((9)(-7) - (3)(-5)) = 0\)

\(2(-5\lambda + 98) + 3(9\lambda - 42) + 5(-63 + 15) = 0\)

\(-10\lambda + 196 + 27\lambda - 126 + 5(-48) = 0\)

\(-10\lambda + 196 + 27\lambda - 126 - 240 = 0\)

\(17\lambda - 170 = 0\)

\(17\lambda = 170\)

\(\lambda = \frac{170}{17} = 10\)

Therefore, the value of \(\lambda\) is 10.